How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice, and we test what it implies against topological alternatives. The law: certifying a feature of spatial scale ε carrying mass fraction f requires polynomial tests of degree N* ≥ log(1/f)/(2ε) — proved via the Chebyshev extremal problem — while a Gauss-quadrature construction gives N* ≥ 4b−1 for a b-scale topology, so the cost is governed by feature fineness, not by feature count. The law is one-sided: it never certifies that a given order suffices, and we exhibit a counterexample (an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H₁ ≠ 0). Its practical content is a calibration rule. The upper-bound construction attains its rate with Gaussian test functions — the RKHS witness of an RBF kernel — so the law predicts which bandwidth an MMD test should use, namely the feature scale. On controlled deformations of real embedding streams we measure σ*/ε with median 1.12 (interquartile range 1.01–1.52, n=26) over three settings and three scales under one protocol, and a bandwidth predicted from a data-driven scale estimate reaches AUC ≥ 0.95. Against an adversary optimised against the defender's own statistics {μ, Σ, k-NN, b}, every one of those statistics is evaded and only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends strongly on choices that are usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR = 1% on a covariance-preserving attack where the first persistence landscape attains 0.00; DTM weighting does not help. Once the summary is chosen sensibly, several findings we would otherwise have reported — degradation with ambient dimension, sensitivity to the reduction dimension, statistically significant domination by kurtosis, failure to alarm in the streaming regime — turn out to be properties of the landscape rather than of persistence. What survives is a cost gap, not a power gap: where persistence works it costs 116× kurtosis, which works at least as well; where kurtosis fails against an adaptive adversary, persistence fails with it. We therefore do not conclude that topological summaries are useless, but that on this task they are dominated by a kernel test whose bandwidth the law tells you how to set.